ODE
\[ 2 y(x) y''(x)=y(x)^2 (a+b y(x))+y'(x)^2 \] ODE Classification
[[_2nd_order, _missing_x]]
Book solution method
TO DO
Mathematica ✓
cpu = 1.74576 (sec), leaf count = 443
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [-\frac {2 i \sqrt {2} \text {$\#$1} \sqrt {\frac {\frac {2 c_1}{\text {$\#$1}}+\sqrt {a^2-2 b c_1}+a}{\sqrt {a^2-2 b c_1}+a}} \sqrt {\frac {2 c_1}{\text {$\#$1} \left (a-\sqrt {a^2-2 b c_1}\right )}+1} F\left (i \sinh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {c_1}{a+\sqrt {a^2-2 b c_1}}}}{\sqrt {\text {$\#$1}}}\right )|\frac {a+\sqrt {a^2-2 b c_1}}{a-\sqrt {a^2-2 b c_1}}\right )}{\sqrt {\frac {c_1}{\sqrt {a^2-2 b c_1}+a}} \sqrt {2 \text {$\#$1}^2 b+4 \left (\text {$\#$1} a+c_1\right )}}\& \right ]\left [c_2+x\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [\frac {2 i \sqrt {2} \text {$\#$1} \sqrt {\frac {\frac {2 c_1}{\text {$\#$1}}+\sqrt {a^2-2 b c_1}+a}{\sqrt {a^2-2 b c_1}+a}} \sqrt {\frac {2 c_1}{\text {$\#$1} \left (a-\sqrt {a^2-2 b c_1}\right )}+1} F\left (i \sinh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {c_1}{a+\sqrt {a^2-2 b c_1}}}}{\sqrt {\text {$\#$1}}}\right )|\frac {a+\sqrt {a^2-2 b c_1}}{a-\sqrt {a^2-2 b c_1}}\right )}{\sqrt {\frac {c_1}{\sqrt {a^2-2 b c_1}+a}} \sqrt {2 \text {$\#$1}^2 b+4 \left (\text {$\#$1} a+c_1\right )}}\& \right ]\left [c_2+x\right ]\right \}\right \}\]
Maple ✓
cpu = 0.292 (sec), leaf count = 69
\[ \left \{ \int ^{y \left ( x \right ) }\!-{\frac {\sqrt {4}}{2}{\frac {1}{\sqrt { \left ( {\frac {{{\it \_a}}^{2}b}{2}}+{\it \_a}\,a+{\it \_C1} \right ) {\it \_a}}}}}{d{\it \_a}}-x-{\it \_C2}=0,\int ^{y \left ( x \right ) }\!{\frac {\sqrt {4}}{2}{\frac {1}{\sqrt { \left ( {\frac {{{\it \_a}}^{2}b}{2}}+{\it \_a}\,a+{\it \_C1} \right ) {\it \_a}}}}}{d{\it \_a}}-x-{\it \_C2}=0 \right \} \] Mathematica raw input
DSolve[2*y[x]*y''[x] == y[x]^2*(a + b*y[x]) + y'[x]^2,y[x],x]
Mathematica raw output
{{y[x] -> InverseFunction[((-2*I)*Sqrt[2]*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[C[1]/(a + Sqrt[a^2 - 2*b*C[1]])])/Sqrt[#1]], (a + Sqrt[a^2 - 2*b*C[1]])/(a - Sqrt[a^2 - 2*b*C[1]])]*Sqrt[(a + Sqrt[a^2 - 2*b*C[1]] + (2*C[1])/#1)/(a + Sqrt[a^2 - 2*b*C[1]])]*Sqrt[1 + (2*C[1])/((a - Sqrt[a^2 - 2*b*C[1]])*#1)]*#1)/(Sqrt[C[1]/(a + Sqrt[a^2 - 2*b*C[1]])]*Sqrt[2*b*#1^2 + 4*(C[1] + a*#1)]) & ][x + C[2]]}, {y[x] -> InverseFunction[((2*I)*Sqrt[2]*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[C[1]/(a + Sqrt[a^2 - 2*b*C[1]])])/Sqrt[#1]], (a + Sqrt[a^2 - 2*b*C[1]])/(a - Sqrt[a^2 - 2*b*C[1]])]*Sqrt[(a + Sqrt[a^2 - 2*b*C[1]] + (2*C[1])/#1)/(a + Sqrt[a^2 - 2*b*C[1]])]*Sqrt[1 + (2*C[1])/((a - Sqrt[a^2 - 2*b*C[1]])*#1)]*#1)/(Sqrt[C[1]/(a + Sqrt[a^2 - 2*b*C[1]])]*Sqrt[2*b*#1^2 + 4*(C[1] + a*#1)]) & ][x + C[2]]}}
Maple raw input
dsolve(2*y(x)*diff(diff(y(x),x),x) = diff(y(x),x)^2+(a+b*y(x))*y(x)^2, y(x),'implicit')
Maple raw output
Intat(-1/2*4^(1/2)/((1/2*_a^2*b+_a*a+_C1)*_a)^(1/2),_a = y(x))-x-_C2 = 0, Intat(1/2*4^(1/2)/((1/2*_a^2*b+_a*a+_C1)*_a)^(1/2),_a = y(x))-x-_C2 = 0